We adopt the following conventions and notations, similar to those of
Misner, Thorne & Wheeler
(1973).
Units are chosen so that c = 1.
The metric signature is (- , + , + , +). The unperturbed background
spacetime is Robertson-Walker with scale factor
a() expressed in terms of
conformal time. A dot (or
) indicates a conformal
time derivative. The comoving expansion rate is written
()
/ a =
aH. The scale factor obeys the Friedmann equation,

(4.1)

The Robertson-Walker line element is written in the general form using
conformal time and comoving
coordinates xi:

(4.2)

Latin indices (i, j, k, etc.) indicate spatial
components while Greek indices (µ,
,
, etc.) indicate all
four spacetime
components; we assume a coordinate basis for tensors. Summation is
implied by repeated upper and lower indices. The inverse 4-metric
gµ
(such that
gµg =
µ)
is used to raise spacetime indices while the inverse 3-metric
ij
(ijjk
= ki)
is used to raise indices of
3-vectors and tensors. Three-tensors are defined in the spatial
hypersurfaces of constant
with metric
ij
and they shall
be clearly distinguished from the spatial components of 4-tensors. We
shall see as we go along how this "3+1 splitting" of spacetime works
when there are metric perturbations.

Many different spatial coordinate systems may be used to cover a
uniform-curvature 3-space. For example, there exist quasi-Cartesian
coordinates (x, y, z) in terms of which the
3-metric components are

(4.3)

We shall use 3-tensor notation to avoid restricting ourselves to any
particular spatial coordinate system. Three-scalars, vectors, and
tensors are invariant under transformations of the spatial coordinate
system in the background spacetime (e.g., rotations). A 3-vector may
be written
A = Aiei where
ei is a basis 3-vector
obeying the dot product rule ei.ej =
ij.
A second-rank 3-tensor may be written (using dyadic notation and the
tensor product) h = hijeiej. We write the
spatial gradient 3-vector operator
=
eii
(i /
xi)
where ei.ej =
ji.
The experts will recognize ei as a
basis one-form but we can treat it as a 3-vector
ei =
ijej because of the isomorphism between vectors
and one-forms.
Because the basis 3-vectors in general have nonvanishing gradients, we
define the covariant derivative (3-gradient) operator
i with
ijk
= 0. If the space is flat (K = 0) and we use
Cartesian coordinates, then
ij
= ij,
i =
i, and
the 3-tensor index notation reduces to elementary
Cartesian notation. If
K 0, the 3-tensor
equations will continue
to look like those in flat space (that is why we use a 3+1 splitting
of spacetime!) except that occasionally terms proportional to K will
appear in our equations.

Our application is not restricted to a flat Robertson-Walker background
but allows for nonzero spatial curvature. This complicates matters
for two reasons. First, we cannot assume Cartesian coordinates. As a
result, for example, the Laplacian of a scalar and the divergence and
curl of a 3-vector involve the determinant of the spatial metric,
det{ij}:

(4.4)

where
ijk =
-1/2
[ijk] is the three-dimensional
Levi-Civita tensor, with [ijk] = + 1 if {ijk} is an even
permutation of {123}, [ijk] = - 1 for an odd permutation, and 0
if any two indices are equal. The factor
-1/2
ensures that
ijk
transforms like a tensor; as an exercise one can show that
ijk =
1/2
[ijk].

The second complication for
K 0 is that gradients
do not commute when
applied to 3-vectors and 3-tensors (though they do commute for 3-scalars).
The basic results are

(4.5)

where [j,
k]
(jk -
kj). The
commutator involves the spatial Riemann tensor, which
for a uniform-curvature space with 3-metric
ij
is simply

(4.6)

Finally, we shall need the evolution equations for the full spacetime metric
gµ.
These are given by the Einstein equations,

(4.7)

where Tµ is the stress-energy tensor and
Gµ is
the Einstein tensor, related to the spacetime Ricci tensor
Rµ by

(4.8)

The spacetime Riemann tensor is defined according to the convention

(4.9)

where the affine connection coefficients are

(4.10)

We see that the Einstein tensor involves second derivatives of the metric
tensor components, so that eq. (4.7) provides second-order
partial differential equations for
gµ.

The reader who is not completely comfortable with the material summarized
above may wish to consult an introductory general relativity textbook, e.g.
Schutz (1985).